Whodunit?

Question

The Threedie brothers, Al, Bert, Chuck, Dick and Eddie, lived in a cabin 3 miles up the old mountain trail, and it was known they didn't get along all that well. This morning, Eddie was found dead behind the cabin, and his brothers, the only suspects in the case, were being questioned by Inspector Sherlock. It was known that, of the four, at least 3 were absolute truth-tellers, and none of them ever lied and told the truth in a single day. All four, of course, denied murdering their brother.

The Inspector started by asking each brother what he had done that morning:

Al: I was analyzing random groups of 3 numbers, and I found that if the numbers sum to zero then their product is the average of their cubes.

Bert: I was analyzing random polygons with 3 sides, and I found that if I trisected all their angles I could make an equilateral triangle.

Chuck: I planted a dozen apple trees out in the orchard, and I found a way to make eighteen rows of 3 trees, each row being dead-on straight.

Dick: I went out and ran 3 miles in the woods, and I've figured out that one of my 3 (living) brothers is lying.

The Inspector called in these clues to one of his friends at BrainDen, and in 3 shakes of a lamb's tail the case was solved. The sound you hear is your phone ringing. It's your chance to be famous!

3. I see rocdocmac's drawing and I wonder if some of those three points are actually collinear.@bonanova Can any 4 points be collinear? If not, Chuck gets my accusation. If they can be, I'll accuse Dick.

Chuck's statement will hold if it can be shown that it's possible to plant 12 trees in such a way that you get 18 straight rows with 3 trees each, the culprit would be Dick. If not proven, then Chuck is the liar and Dick is safe.

I think Chuck is the liar

Edited January 4 by rocdocmac

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It is possible to plant 12 trees in a maximum of 19 rows with each row containing 3 trees in a straight line. Thus, 18 rows with three trees in line seems feasible, but what would the configuration look like? Still a humdinger! C'mon Chuck, show us to prove you are guiltless or not!

Chuck's statement will hold if it can be shown that it's possible to plant 12 trees in such a way that you get 18 straight rows with 3 trees each, the culprit would be Dick. If not proven, then Chuck is the liar and Dick is safe.

I think Chuck is the liar

Regarding (2), can you make a sketch for the Inspector?

Spoiler

He tried, and failed, finding none of the trisectors to be collinear

7 hours ago, Thalia said:

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If I understand right, Bert's statement doesn't hold up starting with an equilateral triangle. But does making a false statement necessarily mean guilt or could it be a mistake on their part?

"Absolute truth-teller" includes "without mistakes," and he did say random. Remember also that ...

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Not the most elegant sketch, but here's one way to plant 12 trees in 18 rows of 3 each.

Thus, if Al and Chuck told the truth and Bert tried (but failed) to make a sketch for the inspector (none of the trisectors colinear), then Dick is the third truth teller. So far all four (living) brothers already got the blame, but it now appears that Bert is in trouble! Bert also mentioned that he "was analyzing random polygons with 3 sides" ... why didn't he say triangles?

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Not the most elegant sketch, but here's one way to plant 12 trees in 18 rows of 3 each.

Thus, if Al and Chuck told the truth and Bert tried (but failed) to make a sketch for the inspector (none of the trisectors colinear), then Dick is the third truth teller. So far all four (living) brothers already got the blame, but it now appears that Bert is in trouble! Bert also mentioned that he "was analyzing random polygons with 3 sides" ... why didn't he say triangles?

I thought that was weird at first too but I attributed it to the Threedie Bros obsession with 3's. I found a picture of a trisected equilateral triangle. None of the triangles with a corner at the trisected angles work because they're only 20 or 40 degrees at the corner. There's 6 smaller more or less identical triangles around the center. I haven't worked out the math but one of the angles is nearly 90 degrees so those are out too. So starting with an equilateral triangle seems to disprove Bert's statement unless he has unusual luck in picking random triangles or you count the original triangle.

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Bert's statement referring to "make an equilateral triangle" (i.e the first Morley triangle) is only partially true. If all of the trisectors are intersected, one can obtain more than one equilateral triangle. Dick may then be telling the truth and refer to Bert as the lying brother. Probably not the answer yet!

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Bert's statement referring to "make an equilateral triangle" (i.e the first Morley triangle) is only partially true. If all of the trisectors are intersected, one can obtain more than one equilateral triangle. Dick may then be telling the truth and refer to Bert as the lying brother. Probably not the answer yet!

Spoiler

If Bert could make more than one, he must also be able to make just one.

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Well, if all of them are telling the truth except for Dick, it must be Dick. Both of his statements could be false, while everyone else has given a truthful statement. Therefore, Dick must be lying, which means he is the only one who could be a liar.

Incidentally - "At least" three of the brothers tell the truth 100%. Could this mean that all of them are telling the truth, and Eddie just committed suicide?